THE SPIRAL ORBIT 



in 



CELESTIAL MECHANICS 

by 



J. G. A. GOEDHART 



"Celestial Orbits are NOT Ellipses, nor can 

THEY EVER BE PARABOLES OR HYPERBOLES. " 






INTRODUCTION 



To 



•i 13/ 



: 






Analytical Mathematical 
Astronomy (A.M.A.) 

By 
J. G. A. GOEDHART 

Officer of the Royal Netherlands Navy, Retired 



H. Poincare — "Les methodes nouvelles de la Mecanique Celeste' 
Paris 1892. Page 1. 



'Le but final de la mecanique celeste est de resoudre cette 
grande question de s avoir si la hi de Newton explique 
a elle settle tous les phenomenes astronomiques. " 



Copyrighted by THE AUTHOR 



ALL RIGHTS RESERVED 
NO BOOK AUTHENTIC WITHOUT MY SIGNATURE. 



New York, March 16, 1921. 



/ 



7k 










PREFACE 



L. S. 

In this brochure I intend to introduce the new and 
original "Analytical Mathematical Astronomy" which 
is greatly different from "Contemporary Astronomy". 

In what way is it different? 

Contemporary Astronomy is based upon: 

First: The Law of Gravitation of Newton. 
Second: The Three Laws of Kepler. 

These Laws of Kepler have been obtained by him 
in an experimental way. Kepler took the different 
results of the observations and computations which 
already had been made by him and other astronomers 
and compared them until he at last arrived at the 
conviction, that certain obtained relations were the 
actual truth. He then incorporated his discoveries 
in the so called Three Laws of Kepler, and presented 
them to the world. 

The scientific world accepted these laws; and these, 
with Newton's Law of Gravitation, formed the foun- 
dation of mathematical astronomy. 

Kepler's manner of investigation, however, was 
merely contemplative and superficial. He considered 
the phenomena in the solar system from the outside, 
without seeing any of the deeper questions, neither 



was there any possibility of grasping former situations, 
nor was there any indication of the origins of those 
phenomena. 

• Analytical Mathematical Astronomy is based 
upon a quite contrary manner of investigation. I 
have attacked the problems connected with the same 
phenomena, which interested Kepler so much, from 
the opposite direction. I have attacked them in the 
analytical way. I started my research from their 
very origins, and have been rewarded by a startling 
abundance of revelations. 

May the contents of these pages, which only claim 
to give an introduction to the extensive revolutionary 
work that I have ready for publication, awaken the 
interest of the readers. 

March 16, 1921 

16628 Endora Road, 
Cleveland, Ohio. 

J. G. A. GOEDHART, 

(From Amsterdam, Holland) 



The Analytical Mathematical 
Astronomy 

"But above all things Truth beareth away the Victory" 

— Inscription New York Public Library 



Concise Historical Review 

THE CIRCULAR CURVE IN CELESTIAL ORBITS 

Different conceptions of the organization of our solar 
system have been evolved during the last centuries of 
history; whence the following theories: 

PTOLEMY, Egyptian Astronomer, about 200 A. D. 
taught the theory that: 

The earth is in the centre of a system of eight large 
hollow spheres, on the surface of each is revolving a 
planet in a circular curve with the earth in its centre. 

COPERNICUS, a Prussian priest, about 1550 A. D. 

advanced another theory: 

The sun is the centre of the solar system, the planets 
revolving around the sun in circular curves, however, 
the sun being placed slightly excentric, 



6 Analytical Mathematical Astronomy 

THE ELLIPTICAL CURVE IN CELESTIAL ORBITS 

KEPLER, a Bavarian astronomer, issued his three 
laws about 1620 and contended: 

The planets, comets, etc. revolve in elliptical orbits 
around the sun, which is placed in one of the foci. 



THE PARABOLIC AND HYPERBOLIC CURVE 
IN CELESTIAL ORBITS 

SIR ISAAC NEWTON, an English mathematician, 
discovered the Law of Gravitation in 1687 : 

Gave the mathematical proof of the three laws of 
Kepler, and of the theory that in certain circumstances 
the secondary bodies had to move around their central 
celestial bodies, along parabolic or hyperbolic curves. 



THE SPIRAL CURVE IN CELESTIAL ORBITS 

My "Analytical Mathematical Astronomy" in 1921 teaches 
the following theory: 

Secondary celestial bodies revolve in spiral orbits around 
the mutual centre of gravitation of all component bodies 
of a system, be it a solar, a planetary or a stellar system. 



Analytical Mathematical Astronomy 



The Six Principal Laws 

being, with the law of Gravitation of Newton, the foundation 
of this Analytical Mathematical Astronomy (A. M. A.) 



FIRST LAW: (Fundamental) 

to be considered as of the same importance as the 
law of Newton, being its natural partner. 
The squares of the centriful forces ( symbol = Fl ) 
inherent in any moving celestial body, around any 
centre of gravitation, following an undisturbed orbit, 
which always will be a spiral, are inversely propor- 
tional to the fifth powers of the distances, (between 
the centres of gravitation of the system and of the 
secondary body.) 

• ri l Pi . • Po 

Remarks to the FIRST LAW : 

1. — This law is the principal feature of A.M. A., it being the 
missing link in the fundamental knowledge about the move- 
ment in heavenly systems and their structures. 

2. — Now we are enabled to answer the question composed 
by H. Poincare, and used as motto on the title page: 

"Le but final de la mecanique celeste 

est de resoudre cette grande question 

de savoir si la loi de Newton explique 

a elle seule tous les ph6nomenes astronomiques." 

with a very determined: No! by lack of this missing link, 
because this first Law forms the second pillar of the foun- 



Analytical Mathematical Astronomy 

dation of the scientific building, which was staggering upon 
its present half foundation — the first pillar : Newton's Law 
of Gravitation. 

3. — After having used ithis first law in its various appli- 
cations, and after having obtained the results of hitherto 
unsolvable problems, it has been revealed that Newton's 
law is rigorous in the most mathematical sense of the word. 

4. — The combination of these two laws, made it possible to 
obtain the set of five other laws which have to replace the 
hitherto prevailing laws of Kepler. 

5. — The whole A.M.A. is built on the foundation of these 
seven laws of which Newton's Law of Gravitation and the 
above mentioned first law are the fundamental ones. 



SECOND LAW: (Substitute for the 1st Law of Kepler) 

Secondary celestial bodies revolve around the centres of 
gravitation of planetary systems in excentric logarithmic 
spiral orbits, the asymptotes of which are ellipses. 

Remarks to the SECOND LAW: 

1. — The excentricity of these orbits is due to a kind of 
disturbance described and dealt with in A.M. A.; an un- 
disturbed logarithmical spiral orbit is not excentric ; its 
asymptote is a circle. 

2. — To this circle has been given the name in A.M.A. of 
standard orbit and in accord to that, is the name of the 
uniform velocity in this orbit Standard Velocity, and the 
semi-diameter of that orbit has been baptised in A.M.A. 
Standard Distance. 

3. — The equation of the undisturbed, not excentric logarith- 
mical spiral is : 

2® 

4p/ 

p a Vo l/a 3n 
p n = Po'.a 

This equation is explained in A.M.A. 






Analytical Mathematical Astronomy 9 

THIRD LAW: {Substitute for the 2d Law of Kepler) 

The radius vector of any secondary body in its spiral 
orbit sweeps in equal times unequal areas, in the way that 
if eventually the attractive force surpasses the centrifugal 
force, the areas will gradually decrease; in the opposite 
situation they will gradually increase. 

Remarks to the THIRD LAW: 

1. — The more a spiral orbit approaches its asymptote, the 
more the unequality of the areas, swept in equal times, dis- 
appears, and vanishes into equality. 



FOURTH LAW : 

In any undisturbed celestial orbit the 4th powers of the 
sideral revolution times * are proportional to the seventh 
powers of the distances. 



J-o • -"-1 — Po • Pi 



FIFTH LAW: {Substitute for the 3rd Law of Kepler) 

Upon comparing the standard orbits of different 
secondary bodies belonging to the same planetary 
system this law reads: 

The squares of the revolution times of standard orbits be- 
longing to one system, are proportional to the third powers 
of the standard distances. 

T 2 : T x 2 = D 3 : V* 



10 Analytical Mathematical Astronomy 

Remarks to the FIFTH LAW: 

1. — In the event of excentrical asymptotical elliptical orbits, 
near which the planets are very close, this law reads as 
follows : 

The fourth powers of the revolution times of asymptotical ellip- 
tical orbits, belonging to one sytem, are proportional to the 
third powers of the products of major and minor axes. 

V ■ V - (a b ) 3 : (aA) 3 

2. — That this improvement does not mean a "slight" 
difference, but a very "important" difference is demon- 
strated by computation. In the event of the orbit of 
Mercurius for instance, Contemporary Astronomy says 
half major axis = 57,536,434 Kilometer; A.M.A. claims it is 
58,157,000 Kilometer, which makes a difference of 620,566 
Kilometer = nearly double the distance of the earth to 
the moon. 

3.— Still more serious is the effect of this improved law 
upon the elements of the orbits of the comets which can 
not be determined at all. Since the revolution time has 
been determined, this fifth law can only reveal to us the 
product of major and minor axes = 4 a b. 

The excentricity, however, is unknown and moreover is 
very important. How to arrive at the value of thermajor 
axis a ? 

What is given to us as such in contemporary books is 
actually the value = V a b 

What is given in those books as the value of the excentricity 
is obtained by calculating it with this erroneous value of a, 
and is therefore worthless. 

The solution of the problem of the elements of the orbit of 
a comet is not possible, unless there be another means to 
arrive at the value of the excentricity. 



Analytical Mathematical Astronomy 11 

SIXTH LAW: 

Upon comparing the standard orbits of different 
secondary bodies belonging to different planetary 
systems (even "outside" our solar system) this Sixth 
Law reads : 

The squares of revolution times of different standard orbits, 
belonging to different systems, are proportional to the third 
powers of the standard distances, and inversely proportional 
to the masses of the systems. 

T 2 : T? = MiDo 3 : MoD^ 



12 Analytical Mathematical Astronomy 



General Remarks 

Comparing the two fundamental laws: Newton's 
Law and my First Law, we see that, if distances vary- 
in the same ratio, the centrifugal force varies in a faster 
tempo than does the attractive force and in the same 
sense. It is therefore obvious that: if any eventu- 
ality should disturb the existing relation of equilibrium 
between those two forces, the result of such eventu- 
ality is limited to two cases: 

1st Case. — If centrifugal force should by any disturb- 
ing impulse suddenly surpass the attractive force any 
arbitrary amount, then the existing spiral orbit will be 
altered in another spiral orbit, bent outward, with 
increasing distances; both forces would decrease, but 
the centrifugal force would decrease in a faster 
tempo, in the way that after some time both forces 
would gradually become equal, and the orbit would 
vanish into a new asymptotical elliptical orbit. 

2nd Case. — In the opposite situation the orbit would 
be bent inward, distances would gradually decrease, 
both forces would increase, but the centrifugal force 
again in a faster tempo. The result would be the same 
as in the first mentioned case. Other situations do 
not exist, therefore it is obvious that A. M. A. teaches: 
there are no parabolic or hyperbolic orbits possible. 
A. M. A. deals fully with this subject. 



Analytical Mathematical Astronomy 13 

Not only converge all computations of A. M. A. in 
an unshakable system of numerical values of all 
possible ephemeris, but A.M. A. unveils an abundance 
of mysteries and puts right many erroneous ideas, 
and issues various new theories, which at once are: 
composed, elaborated, explained, and mathematically 
proved, so that the theories ask only for the most 
serious criticism in order to be settled. 

I will select out of those various theories three 
examples in order to arouse interest: 

1st — The theory of the origin of satellites which is 
fully confirmed by the computations of the twenty best 
known satellites in our solar system, and which theory 
is so powerful, that it unveils many mysteries of which 
I only mention here: the fast rotation of Phobos 
around Mars; the origin and even the place of origin 
of the rings of Saturn; the sequence of the times of 
birth of the satellites of every planet, etc, etc. 

2nd — The mode of derivation of the number of 
vibrations of the ether in any arbitrary light ray 
from any arbitrary planetary orbit, 
for instance: 

Red light 4 x 10 14 

Yellow light 5 x 10 u } Vibrations in one second. 

Blue light 7 x 10 14 

to be derived from the orbit of Neptune or of Phobos. 



14 Analytical Mathematical Astronomy 

This derivation gives at the same time the mathe- 
matical proof of the theory of LORENTZ, that the 
light rays are originated by the electrons revolving 
aroundCthe atoms. 

And last but not least : this derivation gives the 
mathematical proof of the existence of the ether. 

3rd — Another important feature of A.M. A. is the 
disclosure of the only proper method to arrive at the 
mathematically exact value of the mean density of 
the earth. 



The readers of this brochure will excuse me from 
giving any further revelations at this time concerning 
the remarkable discoveries I have made. 

I only wish to make the positive statement that I 
have discovered and worked out my 

ANALYTICAL MATHEMATICAL ASTRONOMY 

without any co-laborer, and have had no help from 
any existing astronomical literature; on the contrary, 
the teachings therein would only show me paths, 
which I am compelled to consider as not leading 
to the actual truth. 



Analytical Mathematical Astronomy 15 

I claim I am publishing today for the first time in 
history the six new laws of this science, which are 
all perfectly rigorous, and I should be pleased to 
have an opportunity for a public discussion with 
contemporary scientific men of the concerning con- 
nected problems. 

In any place of the world! 

Science is international! 

New York, March 16, 1921, 

Temporary Address : 531 West End Avenue, 

J. G. A. GOEDHART 






m " ■ 



